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August 26, 2017

Continuous Surjection ~ nonexistence | TIFR 2013 problem 26

Question:

True/False?

There exists a continuous surjective map from the complex plane onto the non-zero reals.

Hint:

Search for topological invariants.

Discussion:

Under a continuous function, connected set must go to connected set. The complex plane (\mathbb{C}) is connected.

It's image must be connected.

(\mathbb{R}-0) is not connected.

So the statement is False.

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